When red blood cells are counted using a certain electronic counter, the standard deviation (SD)

of repeated counts of the same blood specimen is about .8% of the true value, and the distribution
of repeated counts is approximately normal. For example, this means that if the true value is
5,000,000 cells/mm3, then the SD is 40,000.
(a) If the true value of the red blood count for a certain specimen is 5,000,000 cells/mm3, what
is the probability that the counter would give a reading between 4,900,000 and 5,100,000?
(b) of the true value of the red blood count for a certain specimen is ì, what is the probability
that the counter would give a reading between .98ì and 1.02ì?
(c) A hospital lab performs counts of many specimens every day. For what percentage of these
specimens does the reported blood count differ from the correct value by 2% or more?
I need help with part C.
part A & B I get it
this Q is from the book and the final answer of C is 1.24% as in the book but I need to find how I reach to this answer
thanks

You subtract 1 from the answer you got in b). You would end up with 0.0124 or 1.24%. This is because the range for b) differs from the mean by 2% or less of the mean.

Another way:
since you are finding 2% or more away from the corrected value, it would be easier if you put the mean as 100, therefore 2%+ away from 100 is 98 and below and 102 and above.
Now you can find the probabilities P(x<98) and P(x>102). Add them up and you get your answer.

Oh boy, I hope you're ready for some clownish calculations! Let's tackle part C and find out the percentage of specimens with a difference of 2% or more from the correct value.

To do this, we need to use some z-score magic. The formula for the z-score is (x - μ) / σ, where x is the observed value, μ is the true value, and σ is the standard deviation.

In this case, we want to find the probability of the counter giving a reading that differs from the true value by 2% or more. So, we need to find the probability of getting a z-score of 0.02 or greater.

Now, the SD is given as 0.8% of the true value, which means the SD is 0.008 times the true value. Let's call this new SD σ_new.

Using part A as an example, the counter reading between 4,900,000 and 5,100,000 would have a z-score of [(4,900,000 - 5,000,000) / (0.008 * 5,000,000)] = -1.25. We want the probability of getting a z-score of 0.02 or greater, so we need to find the probability of getting a z-score less than -1.25.

To do this, we can use a standard normal distribution table or a calculator to find the corresponding cumulative probability. Looking up -1.25 in the table, we get approximately 0.1056. This is the probability of getting a z-score less than -1.25, but we want the probability of getting a z-score of 0.02 or greater. So, we subtract this value from 1 to get 1 - 0.1056 = 0.8944.

But wait, we're not done clowning around yet! This probability represents the chance of the counter giving a reading that differs from the true value by 2% or LESS. We want the probability of the counter giving a reading that differs by 2% or MORE, so we subtract this probability from 1 again to get 1 - 0.8944 = 0.1056.

And voila! We have a probability of 0.1056, which is equivalent to 10.56%. Thus, for 10.56% of the specimens, the reported blood count differs from the correct value by 2% or more.

I hope that answers your question and gave you a good laugh along the way! Remember, always have a dose of humor when dealing with statistics.

To calculate the probability that the reported blood count differs from the correct value by 2% or more, we need to find the area under the normal distribution curve beyond 2 standard deviations from the mean.

Given that the standard deviation (SD) is 0.8% of the true value, and the distribution is approximately normal, we can calculate the probability using the Z-score formula.

The Z-score formula is given by:
Z = (X - μ) / σ

Where:
Z is the Z-score
X is the value we want to find the probability for (in this case, 2% above or below the true value)
μ is the mean of the distribution (in this case, the true value)
σ is the standard deviation of the distribution

In this case, the formula becomes:
Z = (X - 5,000,000) / 40,000

Let's calculate the Z-score for a blood count that differs by 2% above the true value:
Z1 = (1.02 * 5,000,000 - 5,000,000) / 40,000

Now, let's calculate the Z-score for a blood count that differs by 2% below the true value:
Z2 = (0.98 * 5,000,000 - 5,000,000) / 40,000

Using a Z-table or a calculator, we can find the probability corresponding to each Z-score.

The probability that the counter would give a reading between 4,900,000 and 5,100,000 is the sum of these two probabilities, since we want to find the probability for both cases (2% above and 2% below the true value):

P = P(Z1) + P(Z2)

You can use a Z-table or a calculator with a normal distribution function to find the probabilities associated with the Z-scores and add them together.

To solve part C, let's first understand the problem. We need to find the percentage of specimens for which the reported blood count differs from the correct value by 2% or more.

Given that the standard deviation (SD) of repeated counts is 0.8% of the true value, we can use this information to calculate the probability of a count falling outside a certain range.

Step 1: Find the value of 2% of the true count.
For a count of 5,000,000 cells/mm3 (true value), 2% of this value is:
2% * 5,000,000 = 100,000 cells/mm3

Step 2: Calculate the number of standard deviations 100,000 cells/mm3 is from the mean.
The standard deviation is 40,000 cells/mm3.
Number of standard deviations = 100,000 / 40,000 = 2.5

Step 3: Find the probability of a count falling outside 2.5 standard deviations from the mean.
Using a normal distribution table or calculator, we can find the probability of a count falling outside 2.5 standard deviations. This probability corresponds to the area under the curve beyond 2.5 standard deviations.
From the normal distribution table, the probability for 2.5 standard deviations is approximately 0.0062 or 0.62%.

However, we want to find the probability of the count differing by 2% or more, so we need to consider both tails of the normal distribution. We multiply the probability by 2:
0.0062 * 2 = 0.0124 or 1.24%

Therefore, the answer to part C is that for approximately 1.24% of the specimens, the reported blood count differs from the correct value by 2% or more.